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Properties

Label 4-23552-1.1-c1e2-0-0
Degree $4$
Conductor $23552$
Sign $1$
Analytic cond. $1.50169$
Root an. cond. $1.10699$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·9-s + 3·23-s + 2·25-s + 12·31-s + 4·41-s − 14·49-s − 8·71-s − 12·73-s − 4·79-s − 5·81-s + 8·89-s − 8·97-s − 4·103-s + 28·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 6·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 2/3·9-s + 0.625·23-s + 2/5·25-s + 2.15·31-s + 0.624·41-s − 2·49-s − 0.949·71-s − 1.40·73-s − 0.450·79-s − 5/9·81-s + 0.847·89-s − 0.812·97-s − 0.394·103-s + 2.63·113-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.461·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23552 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23552 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(23552\)    =    \(2^{10} \cdot 23\)
Sign: $1$
Analytic conductor: \(1.50169\)
Root analytic conductor: \(1.10699\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 23552,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.259844392\)
\(L(\frac12)\) \(\approx\) \(1.259844392\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
23$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 4 T + p T^{2} ) \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.13.a_g
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
29$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.29.a_aba
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.31.am_dq
37$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.37.a_ac
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.ae_w
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \) 2.43.a_cg
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.53.a_be
59$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.59.a_acw
61$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.61.a_ck
67$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.67.a_ba
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.i_fm
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.m_gk
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.e_gc
83$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.83.a_ba
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.ai_dq
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.97.i_hy
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.69099860939119063949486437790, −10.03256030546938523029884952175, −9.893297447050737779667200345653, −9.080996812119159195501793968112, −8.629145689615009604977670515657, −7.980958205112690661543697137162, −7.47483879931360866022385782664, −6.79226340058756511884859043180, −6.35026227136431346633616438634, −5.62873732211563836507638997949, −4.70982063836783104451929968744, −4.44575763121154858836688963151, −3.37532459823709551349209825769, −2.62503936236777372358374163884, −1.32704857384085566366678620514, 1.32704857384085566366678620514, 2.62503936236777372358374163884, 3.37532459823709551349209825769, 4.44575763121154858836688963151, 4.70982063836783104451929968744, 5.62873732211563836507638997949, 6.35026227136431346633616438634, 6.79226340058756511884859043180, 7.47483879931360866022385782664, 7.980958205112690661543697137162, 8.629145689615009604977670515657, 9.080996812119159195501793968112, 9.893297447050737779667200345653, 10.03256030546938523029884952175, 10.69099860939119063949486437790

Graph of the $Z$-function along the critical line